// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2009 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 2010 Benoit Jacob <jacob.benoit.1@gmail.com>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

#ifndef EIGEN_HOUSEHOLDER_SEQUENCE_H
#define EIGEN_HOUSEHOLDER_SEQUENCE_H

namespace Eigen {

/** \ingroup Householder_Module
  * \householder_module
  * \class HouseholderSequence
  * \brief Sequence of Householder reflections acting on subspaces with decreasing size
  * \tparam VectorsType type of matrix containing the Householder vectors
  * \tparam CoeffsType  type of vector containing the Householder coefficients
  * \tparam Side        either OnTheLeft (the default) or OnTheRight
  *
  * This class represents a product sequence of Householder reflections where the first Householder reflection
  * acts on the whole space, the second Householder reflection leaves the one-dimensional subspace spanned by
  * the first unit vector invariant, the third Householder reflection leaves the two-dimensional subspace
  * spanned by the first two unit vectors invariant, and so on up to the last reflection which leaves all but
  * one dimensions invariant and acts only on the last dimension. Such sequences of Householder reflections
  * are used in several algorithms to zero out certain parts of a matrix. Indeed, the methods
  * HessenbergDecomposition::matrixQ(), Tridiagonalization::matrixQ(), HouseholderQR::householderQ(),
  * and ColPivHouseholderQR::householderQ() all return a %HouseholderSequence.
  *
  * More precisely, the class %HouseholderSequence represents an \f$ n \times n \f$ matrix \f$ H \f$ of the
  * form \f$ H = \prod_{i=0}^{n-1} H_i \f$ where the i-th Householder reflection is \f$ H_i = I - h_i v_i
  * v_i^* \f$. The i-th Householder coefficient \f$ h_i \f$ is a scalar and the i-th Householder vector \f$
  * v_i \f$ is a vector of the form
  * \f[
  * v_i = [\underbrace{0, \ldots, 0}_{i-1\mbox{ zeros}}, 1, \underbrace{*, \ldots,*}_{n-i\mbox{ arbitrary entries}} ].
  * \f]
  * The last \f$ n-i \f$ entries of \f$ v_i \f$ are called the essential part of the Householder vector.
  *
  * Typical usages are listed below, where H is a HouseholderSequence:
  * \code
  * A.applyOnTheRight(H);             // A = A * H
  * A.applyOnTheLeft(H);              // A = H * A
  * A.applyOnTheRight(H.adjoint());   // A = A * H^*
  * A.applyOnTheLeft(H.adjoint());    // A = H^* * A
  * MatrixXd Q = H;                   // conversion to a dense matrix
  * \endcode
  * In addition to the adjoint, you can also apply the inverse (=adjoint), the transpose, and the conjugate operators.
  *
  * See the documentation for HouseholderSequence(const VectorsType&, const CoeffsType&) for an example.
  *
  * \sa MatrixBase::applyOnTheLeft(), MatrixBase::applyOnTheRight()
  */

namespace internal {

    template <typename VectorsType, typename CoeffsType, int Side> struct traits<HouseholderSequence<VectorsType, CoeffsType, Side>>
    {
        typedef typename VectorsType::Scalar Scalar;
        typedef typename VectorsType::StorageIndex StorageIndex;
        typedef typename VectorsType::StorageKind StorageKind;
        enum
        {
            RowsAtCompileTime = Side == OnTheLeft ? traits<VectorsType>::RowsAtCompileTime : traits<VectorsType>::ColsAtCompileTime,
            ColsAtCompileTime = RowsAtCompileTime,
            MaxRowsAtCompileTime = Side == OnTheLeft ? traits<VectorsType>::MaxRowsAtCompileTime : traits<VectorsType>::MaxColsAtCompileTime,
            MaxColsAtCompileTime = MaxRowsAtCompileTime,
            Flags = 0
        };
    };

    struct HouseholderSequenceShape
    {
    };

    template <typename VectorsType, typename CoeffsType, int Side>
    struct evaluator_traits<HouseholderSequence<VectorsType, CoeffsType, Side>>
        : public evaluator_traits_base<HouseholderSequence<VectorsType, CoeffsType, Side>>
    {
        typedef HouseholderSequenceShape Shape;
    };

    template <typename VectorsType, typename CoeffsType, int Side> struct hseq_side_dependent_impl
    {
        typedef Block<const VectorsType, Dynamic, 1> EssentialVectorType;
        typedef HouseholderSequence<VectorsType, CoeffsType, OnTheLeft> HouseholderSequenceType;
        static EIGEN_DEVICE_FUNC inline const EssentialVectorType essentialVector(const HouseholderSequenceType& h, Index k)
        {
            Index start = k + 1 + h.m_shift;
            return Block<const VectorsType, Dynamic, 1>(h.m_vectors, start, k, h.rows() - start, 1);
        }
    };

    template <typename VectorsType, typename CoeffsType> struct hseq_side_dependent_impl<VectorsType, CoeffsType, OnTheRight>
    {
        typedef Transpose<Block<const VectorsType, 1, Dynamic>> EssentialVectorType;
        typedef HouseholderSequence<VectorsType, CoeffsType, OnTheRight> HouseholderSequenceType;
        static inline const EssentialVectorType essentialVector(const HouseholderSequenceType& h, Index k)
        {
            Index start = k + 1 + h.m_shift;
            return Block<const VectorsType, 1, Dynamic>(h.m_vectors, k, start, 1, h.rows() - start).transpose();
        }
    };

    template <typename OtherScalarType, typename MatrixType> struct matrix_type_times_scalar_type
    {
        typedef typename ScalarBinaryOpTraits<OtherScalarType, typename MatrixType::Scalar>::ReturnType ResultScalar;
        typedef Matrix<ResultScalar,
                       MatrixType::RowsAtCompileTime,
                       MatrixType::ColsAtCompileTime,
                       0,
                       MatrixType::MaxRowsAtCompileTime,
                       MatrixType::MaxColsAtCompileTime>
            Type;
    };

}  // end namespace internal

template <typename VectorsType, typename CoeffsType, int Side> class HouseholderSequence : public EigenBase<HouseholderSequence<VectorsType, CoeffsType, Side>>
{
    typedef typename internal::hseq_side_dependent_impl<VectorsType, CoeffsType, Side>::EssentialVectorType EssentialVectorType;

public:
    enum
    {
        RowsAtCompileTime = internal::traits<HouseholderSequence>::RowsAtCompileTime,
        ColsAtCompileTime = internal::traits<HouseholderSequence>::ColsAtCompileTime,
        MaxRowsAtCompileTime = internal::traits<HouseholderSequence>::MaxRowsAtCompileTime,
        MaxColsAtCompileTime = internal::traits<HouseholderSequence>::MaxColsAtCompileTime
    };
    typedef typename internal::traits<HouseholderSequence>::Scalar Scalar;

    typedef HouseholderSequence<
        typename internal::
            conditional<NumTraits<Scalar>::IsComplex, typename internal::remove_all<typename VectorsType::ConjugateReturnType>::type, VectorsType>::type,
        typename internal::
            conditional<NumTraits<Scalar>::IsComplex, typename internal::remove_all<typename CoeffsType::ConjugateReturnType>::type, CoeffsType>::type,
        Side>
        ConjugateReturnType;

    typedef HouseholderSequence<VectorsType,
                                typename internal::conditional<NumTraits<Scalar>::IsComplex,
                                                               typename internal::remove_all<typename CoeffsType::ConjugateReturnType>::type,
                                                               CoeffsType>::type,
                                Side>
        AdjointReturnType;

    typedef HouseholderSequence<typename internal::conditional<NumTraits<Scalar>::IsComplex,
                                                               typename internal::remove_all<typename VectorsType::ConjugateReturnType>::type,
                                                               VectorsType>::type,
                                CoeffsType,
                                Side>
        TransposeReturnType;

    typedef HouseholderSequence<typename internal::add_const<VectorsType>::type, typename internal::add_const<CoeffsType>::type, Side> ConstHouseholderSequence;

    /** \brief Constructor.
      * \param[in]  v      %Matrix containing the essential parts of the Householder vectors
      * \param[in]  h      Vector containing the Householder coefficients
      *
      * Constructs the Householder sequence with coefficients given by \p h and vectors given by \p v. The
      * i-th Householder coefficient \f$ h_i \f$ is given by \p h(i) and the essential part of the i-th
      * Householder vector \f$ v_i \f$ is given by \p v(k,i) with \p k > \p i (the subdiagonal part of the
      * i-th column). If \p v has fewer columns than rows, then the Householder sequence contains as many
      * Householder reflections as there are columns.
      *
      * \note The %HouseholderSequence object stores \p v and \p h by reference.
      *
      * Example: \include HouseholderSequence_HouseholderSequence.cpp
      * Output: \verbinclude HouseholderSequence_HouseholderSequence.out
      *
      * \sa setLength(), setShift()
      */
    EIGEN_DEVICE_FUNC
    HouseholderSequence(const VectorsType& v, const CoeffsType& h) : m_vectors(v), m_coeffs(h), m_reverse(false), m_length(v.diagonalSize()), m_shift(0) {}

    /** \brief Copy constructor. */
    EIGEN_DEVICE_FUNC
    HouseholderSequence(const HouseholderSequence& other)
        : m_vectors(other.m_vectors), m_coeffs(other.m_coeffs), m_reverse(other.m_reverse), m_length(other.m_length), m_shift(other.m_shift)
    {
    }

    /** \brief Number of rows of transformation viewed as a matrix.
      * \returns Number of rows
      * \details This equals the dimension of the space that the transformation acts on.
      */
    EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR Index rows() const EIGEN_NOEXCEPT { return Side == OnTheLeft ? m_vectors.rows() : m_vectors.cols(); }

    /** \brief Number of columns of transformation viewed as a matrix.
      * \returns Number of columns
      * \details This equals the dimension of the space that the transformation acts on.
      */
    EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR Index cols() const EIGEN_NOEXCEPT { return rows(); }

    /** \brief Essential part of a Householder vector.
      * \param[in]  k  Index of Householder reflection
      * \returns    Vector containing non-trivial entries of k-th Householder vector
      *
      * This function returns the essential part of the Householder vector \f$ v_i \f$. This is a vector of
      * length \f$ n-i \f$ containing the last \f$ n-i \f$ entries of the vector
      * \f[
      * v_i = [\underbrace{0, \ldots, 0}_{i-1\mbox{ zeros}}, 1, \underbrace{*, \ldots,*}_{n-i\mbox{ arbitrary entries}} ].
      * \f]
      * The index \f$ i \f$ equals \p k + shift(), corresponding to the k-th column of the matrix \p v
      * passed to the constructor.
      *
      * \sa setShift(), shift()
      */
    EIGEN_DEVICE_FUNC
    const EssentialVectorType essentialVector(Index k) const
    {
        eigen_assert(k >= 0 && k < m_length);
        return internal::hseq_side_dependent_impl<VectorsType, CoeffsType, Side>::essentialVector(*this, k);
    }

    /** \brief %Transpose of the Householder sequence. */
    TransposeReturnType transpose() const
    {
        return TransposeReturnType(m_vectors.conjugate(), m_coeffs).setReverseFlag(!m_reverse).setLength(m_length).setShift(m_shift);
    }

    /** \brief Complex conjugate of the Householder sequence. */
    ConjugateReturnType conjugate() const
    {
        return ConjugateReturnType(m_vectors.conjugate(), m_coeffs.conjugate()).setReverseFlag(m_reverse).setLength(m_length).setShift(m_shift);
    }

    /** \returns an expression of the complex conjugate of \c *this if Cond==true,
     *           returns \c *this otherwise.
     */
    template <bool Cond> EIGEN_DEVICE_FUNC inline typename internal::conditional<Cond, ConjugateReturnType, ConstHouseholderSequence>::type conjugateIf() const
    {
        typedef typename internal::conditional<Cond, ConjugateReturnType, ConstHouseholderSequence>::type ReturnType;
        return ReturnType(m_vectors.template conjugateIf<Cond>(), m_coeffs.template conjugateIf<Cond>());
    }

    /** \brief Adjoint (conjugate transpose) of the Householder sequence. */
    AdjointReturnType adjoint() const
    {
        return AdjointReturnType(m_vectors, m_coeffs.conjugate()).setReverseFlag(!m_reverse).setLength(m_length).setShift(m_shift);
    }

    /** \brief Inverse of the Householder sequence (equals the adjoint). */
    AdjointReturnType inverse() const { return adjoint(); }

    /** \internal */
    template <typename DestType> inline EIGEN_DEVICE_FUNC void evalTo(DestType& dst) const
    {
        Matrix<Scalar, DestType::RowsAtCompileTime, 1, AutoAlign | ColMajor, DestType::MaxRowsAtCompileTime, 1> workspace(rows());
        evalTo(dst, workspace);
    }

    /** \internal */
    template <typename Dest, typename Workspace> EIGEN_DEVICE_FUNC void evalTo(Dest& dst, Workspace& workspace) const
    {
        workspace.resize(rows());
        Index vecs = m_length;
        if (internal::is_same_dense(dst, m_vectors))
        {
            // in-place
            dst.diagonal().setOnes();
            dst.template triangularView<StrictlyUpper>().setZero();
            for (Index k = vecs - 1; k >= 0; --k)
            {
                Index cornerSize = rows() - k - m_shift;
                if (m_reverse)
                    dst.bottomRightCorner(cornerSize, cornerSize).applyHouseholderOnTheRight(essentialVector(k), m_coeffs.coeff(k), workspace.data());
                else
                    dst.bottomRightCorner(cornerSize, cornerSize).applyHouseholderOnTheLeft(essentialVector(k), m_coeffs.coeff(k), workspace.data());

                // clear the off diagonal vector
                dst.col(k).tail(rows() - k - 1).setZero();
            }
            // clear the remaining columns if needed
            for (Index k = 0; k < cols() - vecs; ++k) dst.col(k).tail(rows() - k - 1).setZero();
        }
        else if (m_length > BlockSize)
        {
            dst.setIdentity(rows(), rows());
            if (m_reverse)
                applyThisOnTheLeft(dst, workspace, true);
            else
                applyThisOnTheLeft(dst, workspace, true);
        }
        else
        {
            dst.setIdentity(rows(), rows());
            for (Index k = vecs - 1; k >= 0; --k)
            {
                Index cornerSize = rows() - k - m_shift;
                if (m_reverse)
                    dst.bottomRightCorner(cornerSize, cornerSize).applyHouseholderOnTheRight(essentialVector(k), m_coeffs.coeff(k), workspace.data());
                else
                    dst.bottomRightCorner(cornerSize, cornerSize).applyHouseholderOnTheLeft(essentialVector(k), m_coeffs.coeff(k), workspace.data());
            }
        }
    }

    /** \internal */
    template <typename Dest> inline void applyThisOnTheRight(Dest& dst) const
    {
        Matrix<Scalar, 1, Dest::RowsAtCompileTime, RowMajor, 1, Dest::MaxRowsAtCompileTime> workspace(dst.rows());
        applyThisOnTheRight(dst, workspace);
    }

    /** \internal */
    template <typename Dest, typename Workspace> inline void applyThisOnTheRight(Dest& dst, Workspace& workspace) const
    {
        workspace.resize(dst.rows());
        for (Index k = 0; k < m_length; ++k)
        {
            Index actual_k = m_reverse ? m_length - k - 1 : k;
            dst.rightCols(rows() - m_shift - actual_k).applyHouseholderOnTheRight(essentialVector(actual_k), m_coeffs.coeff(actual_k), workspace.data());
        }
    }

    /** \internal */
    template <typename Dest> inline void applyThisOnTheLeft(Dest& dst, bool inputIsIdentity = false) const
    {
        Matrix<Scalar, 1, Dest::ColsAtCompileTime, RowMajor, 1, Dest::MaxColsAtCompileTime> workspace;
        applyThisOnTheLeft(dst, workspace, inputIsIdentity);
    }

    /** \internal */
    template <typename Dest, typename Workspace> inline void applyThisOnTheLeft(Dest& dst, Workspace& workspace, bool inputIsIdentity = false) const
    {
        if (inputIsIdentity && m_reverse)
            inputIsIdentity = false;
        // if the entries are large enough, then apply the reflectors by block
        if (m_length >= BlockSize && dst.cols() > 1)
        {
            // Make sure we have at least 2 useful blocks, otherwise it is point-less:
            Index blockSize = m_length < Index(2 * BlockSize) ? (m_length + 1) / 2 : Index(BlockSize);
            for (Index i = 0; i < m_length; i += blockSize)
            {
                Index end = m_reverse ? (std::min)(m_length, i + blockSize) : m_length - i;
                Index k = m_reverse ? i : (std::max)(Index(0), end - blockSize);
                Index bs = end - k;
                Index start = k + m_shift;

                typedef Block<typename internal::remove_all<VectorsType>::type, Dynamic, Dynamic> SubVectorsType;
                SubVectorsType sub_vecs1(m_vectors.const_cast_derived(),
                                         Side == OnTheRight ? k : start,
                                         Side == OnTheRight ? start : k,
                                         Side == OnTheRight ? bs : m_vectors.rows() - start,
                                         Side == OnTheRight ? m_vectors.cols() - start : bs);
                typename internal::conditional<Side == OnTheRight, Transpose<SubVectorsType>, SubVectorsType&>::type sub_vecs(sub_vecs1);

                Index dstStart = dst.rows() - rows() + m_shift + k;
                Index dstRows = rows() - m_shift - k;
                Block<Dest, Dynamic, Dynamic> sub_dst(dst, dstStart, inputIsIdentity ? dstStart : 0, dstRows, inputIsIdentity ? dstRows : dst.cols());
                apply_block_householder_on_the_left(sub_dst, sub_vecs, m_coeffs.segment(k, bs), !m_reverse);
            }
        }
        else
        {
            workspace.resize(dst.cols());
            for (Index k = 0; k < m_length; ++k)
            {
                Index actual_k = m_reverse ? k : m_length - k - 1;
                Index dstStart = rows() - m_shift - actual_k;
                dst.bottomRightCorner(dstStart, inputIsIdentity ? dstStart : dst.cols())
                    .applyHouseholderOnTheLeft(essentialVector(actual_k), m_coeffs.coeff(actual_k), workspace.data());
            }
        }
    }

    /** \brief Computes the product of a Householder sequence with a matrix.
      * \param[in]  other  %Matrix being multiplied.
      * \returns    Expression object representing the product.
      *
      * This function computes \f$ HM \f$ where \f$ H \f$ is the Householder sequence represented by \p *this
      * and \f$ M \f$ is the matrix \p other.
      */
    template <typename OtherDerived>
    typename internal::matrix_type_times_scalar_type<Scalar, OtherDerived>::Type operator*(const MatrixBase<OtherDerived>& other) const
    {
        typename internal::matrix_type_times_scalar_type<Scalar, OtherDerived>::Type res(
            other.template cast<typename internal::matrix_type_times_scalar_type<Scalar, OtherDerived>::ResultScalar>());
        applyThisOnTheLeft(res, internal::is_identity<OtherDerived>::value && res.rows() == res.cols());
        return res;
    }

    template <typename _VectorsType, typename _CoeffsType, int _Side> friend struct internal::hseq_side_dependent_impl;

    /** \brief Sets the length of the Householder sequence.
      * \param [in]  length  New value for the length.
      *
      * By default, the length \f$ n \f$ of the Householder sequence \f$ H = H_0 H_1 \ldots H_{n-1} \f$ is set
      * to the number of columns of the matrix \p v passed to the constructor, or the number of rows if that
      * is smaller. After this function is called, the length equals \p length.
      *
      * \sa length()
      */
    EIGEN_DEVICE_FUNC
    HouseholderSequence& setLength(Index length)
    {
        m_length = length;
        return *this;
    }

    /** \brief Sets the shift of the Householder sequence.
      * \param [in]  shift  New value for the shift.
      *
      * By default, a %HouseholderSequence object represents \f$ H = H_0 H_1 \ldots H_{n-1} \f$ and the i-th
      * column of the matrix \p v passed to the constructor corresponds to the i-th Householder
      * reflection. After this function is called, the object represents \f$ H = H_{\mathrm{shift}}
      * H_{\mathrm{shift}+1} \ldots H_{n-1} \f$ and the i-th column of \p v corresponds to the (shift+i)-th
      * Householder reflection.
      *
      * \sa shift()
      */
    EIGEN_DEVICE_FUNC
    HouseholderSequence& setShift(Index shift)
    {
        m_shift = shift;
        return *this;
    }

    EIGEN_DEVICE_FUNC
    Index length() const { return m_length; } /**< \brief Returns the length of the Householder sequence. */

    EIGEN_DEVICE_FUNC
    Index shift() const { return m_shift; } /**< \brief Returns the shift of the Householder sequence. */

    /* Necessary for .adjoint() and .conjugate() */
    template <typename VectorsType2, typename CoeffsType2, int Side2> friend class HouseholderSequence;

protected:
    /** \internal
      * \brief Sets the reverse flag.
      * \param [in]  reverse  New value of the reverse flag.
      *
      * By default, the reverse flag is not set. If the reverse flag is set, then this object represents
      * \f$ H^r = H_{n-1} \ldots H_1 H_0 \f$ instead of \f$ H = H_0 H_1 \ldots H_{n-1} \f$.
      * \note For real valued HouseholderSequence this is equivalent to transposing \f$ H \f$.
      *
      * \sa reverseFlag(), transpose(), adjoint()
      */
    HouseholderSequence& setReverseFlag(bool reverse)
    {
        m_reverse = reverse;
        return *this;
    }

    bool reverseFlag() const { return m_reverse; } /**< \internal \brief Returns the reverse flag. */

    typename VectorsType::Nested m_vectors;
    typename CoeffsType::Nested m_coeffs;
    bool m_reverse;
    Index m_length;
    Index m_shift;
    enum
    {
        BlockSize = 48
    };
};

/** \brief Computes the product of a matrix with a Householder sequence.
  * \param[in]  other  %Matrix being multiplied.
  * \param[in]  h      %HouseholderSequence being multiplied.
  * \returns    Expression object representing the product.
  *
  * This function computes \f$ MH \f$ where \f$ M \f$ is the matrix \p other and \f$ H \f$ is the
  * Householder sequence represented by \p h.
  */
template <typename OtherDerived, typename VectorsType, typename CoeffsType, int Side>
typename internal::matrix_type_times_scalar_type<typename VectorsType::Scalar, OtherDerived>::Type
operator*(const MatrixBase<OtherDerived>& other, const HouseholderSequence<VectorsType, CoeffsType, Side>& h)
{
    typename internal::matrix_type_times_scalar_type<typename VectorsType::Scalar, OtherDerived>::Type res(
        other.template cast<typename internal::matrix_type_times_scalar_type<typename VectorsType::Scalar, OtherDerived>::ResultScalar>());
    h.applyThisOnTheRight(res);
    return res;
}

/** \ingroup Householder_Module \householder_module
  * \brief Convenience function for constructing a Householder sequence.
  * \returns A HouseholderSequence constructed from the specified arguments.
  */
template <typename VectorsType, typename CoeffsType> HouseholderSequence<VectorsType, CoeffsType> householderSequence(const VectorsType& v, const CoeffsType& h)
{
    return HouseholderSequence<VectorsType, CoeffsType, OnTheLeft>(v, h);
}

/** \ingroup Householder_Module \householder_module
  * \brief Convenience function for constructing a Householder sequence.
  * \returns A HouseholderSequence constructed from the specified arguments.
  * \details This function differs from householderSequence() in that the template argument \p OnTheSide of
  * the constructed HouseholderSequence is set to OnTheRight, instead of the default OnTheLeft.
  */
template <typename VectorsType, typename CoeffsType>
HouseholderSequence<VectorsType, CoeffsType, OnTheRight> rightHouseholderSequence(const VectorsType& v, const CoeffsType& h)
{
    return HouseholderSequence<VectorsType, CoeffsType, OnTheRight>(v, h);
}

}  // end namespace Eigen

#endif  // EIGEN_HOUSEHOLDER_SEQUENCE_H
